M. Fried, R. Guralnick and J. Saxl, Schur covers and Carlitz's conjecture, Israel J. 82 (1993), 157--225.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....bijections on rational points: Theorem 1. If f is exceptional, then f maps X(F q ) bijectively onto Y (F q ) This result is due to Lenstra (unpublished) Special cases and weaker versions were previously proved by Davenport and Lewis [DL] MacCluer [Mac] Williams [Wi] Cohen [Co] and Fried [Fr, Fr2, FGS]. See [LMZ] for variants of this result over infinite constant fields. Note that, if f is exceptional over F q , then f is also exceptional over F q n for infinitely many n. Thus, f induces a bijection X(F q n ) Y (F q n ) for infinitely many n. This unusual property is the most ....

M. Fried, R. Guralnick and J. Saxl, Schur covers and Carlitz's conjecture, Israel J. Math. 82 (1993), 157--225.

....if is large compared to the degree of f . Further, exceptional polynomials can be characterized as those polynomials which are permutation polynomials over infinitely many finite extensions of F ; for proofs of these and other statements about exceptional polynomials in this introduction see [5]. The composition of two exceptional polynomials is itself exceptional, and conversely the composition factors of an exceptional polynomial are also exceptional; so one is interested in classifying the indecomposable exceptional polynomials. Some simple types of indecomposable exceptional ....

....point. Thus, a polynomial is exceptional if and only if the action of its monodromy groups is exceptional. Finally, we note that the above definitions work just as well for rational functions as for polynomials, and the same basic results hold in that context. Recently Fried, Guralnick, and Saxl [5] used these Galois theoretic correspondences to prove severe restrictions on the possible monodromy groups of an indecomposable exceptional polynomial. The bulk of their effort was group theoretic: they used the above conditions on the monodromy groups, together with another condition reflecting ....

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M. D. Fried, R. M. Guralnick, and J. Saxl, Schur covers and Carlitz's conjecture, Israel J. Math. 82 (1993), 157--225.

....A primer on exceptional polynomials is included as an appendix to this paper. Over the years numerous authors have contributed to the theory of exceptional polynomials, with steady success, but a radical change in perspective came in 1993. This was due to the work of Fried, Guralnick and Saxl [9], who used hard group theory (including the classification of finite simple groups) in order to severely restrict the possibilities for the Galois group Gal(f(X) Gamma t; k(t) of an exceptional polynomial. Their work provided hope for a complete classification of exceptional polynomials, ....

....that the Galois group is typically an affine group (that is, a group of invertible affine transformations of a vector space) except for certain unexpected possibilities over fields of characteristic two and three. Every exceptional polynomial known in 1993 had affine Galois group; but following [9] there was a flurry of activity which saw the construction of new (non affine) exceptional polynomials in characteristics two and three. In fact, in recent work Guralnick and I have completely classified the non affine exceptional polynomials [13] However, for the non affine exceptional ....

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M. D. Fried, R. M. Guralnick, and J. Saxl, Schur covers and Carlitz's conjecture, Israel J. Math. 82 (1993), 157--225.

....Yet, the classification of affine groups arising as monodromy groups of exceptional polynomials over a finite field is incomplete. 23, x5 x6] discusses this and details on the genus 0 problem. Exceptional polynomials over F q give one one maps on F q t for infinitely many t. The main result of [20] is that excluding a known list, indecomposable exceptional polynomials over F q have these properties. ffl They have degree p u for some u. ffl Their monodromy group is an affine group acting on F p u . 2.1. What to expect of monodromy groups from genus 0 covers. Let f 2 C (x) be a rational ....

M. Fried, R. Guralnick and J. Saxl, Schur covers and Carlitz's conjecture, Israel J. 82 (1993), 157--225.

....Davenport s problem in characteristic p exemplifies this difference. 5.1. Simple groups as a resource for investigations into affine groups. Specifically, we hope this progress on Davenport s problem inspires completing the description of exceptional polynomial (and rational function) covers. [FGS93] reduced this latter problem (using the classification) to investigating arithmetic geometric monodromy group pairs ( G; G) with G = V Theta s H (affine group) and H 22 M. FRIED acting irreducibly on a vector space V , of order p r . Recall: A polynomial f (or rational function) is ....

....are subgroups of the wreath product of the monodromy groups of the composition factors. This reduced the problem to where the monodromy group is primitive. GT90] then applied the classification based taxonomy of primitive groups by Aschbacher O Nan Scott [AS85] into five types; see exposition of [FGS93, x13]) Further, they did several primitive cases. One is of affine groups (x5.1) V Theta s H with H acting irreducibly on the vector space V . Neu93] improved the results of [GT90] in the affine case. Mag93] handled the case of primitive groups arising from sporadic simple groups. A result from ....

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M. Fried, R. Guralnick and J. Saxl, Schur covers and Carlitz's conjecture, Israel J. 82 (1993), 157--225.

....monodromy group of f p over K p respectively. Let B be the subgroup of A such that B=G induces the decomposition group of a place of K lying above p. Then, if C 0 was big enough, we have the following result by Fried [Fri74, Lemma 1] A p = B G p = G: Furthermore, it is known (see [FGS93] that there is a bound C 00 such that if jK p j C 00 , then f p is bijective on P 1 (K p ) if and only if (A p ; G p ) is exceptional. This proves (b) and the only if part of (a) To prove the if part of (a) assume that there is B as in (a) Let bG generate A=G. By Chebotarev s ....

....on P 1 (F q m) for infinitely many m 2 N. Let A and G be the arithmetic and geometric monodromy group of f . Then A=G is cyclic, and f is exceptional if and only if (A; G) is exceptional with respect to the action on the roots of f(X) Gamma t, this follows from proof of Theorem 2.10. 12 In [FGS93] the possible monodromy groups of exceptional polynomials have been classified, and the question about existence of actual examples has been answered positively for all non affine groups in [CM94] LZ96] and for some affine groups in [GM97] There is little hope to achieve an analogous result ....

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M. Fried, R. Guralnick, J. Saxl, Schur covers and Carlitz's conjecture, Israel J. Math. (1993), 82, 157--225.

....that the constant in the above error term depends only on n, not on q. Thus, to understand the asymptotic behavior of V f , it suffices to understand the quotient s 0 = jS 0 j jN j : It is clear that s 0 0 with equality holding if and only if f(T ) is an exceptional polynomial over F q (see [FGS] for a classification of the possible monodromy groups for exceptional polynomials) Lenstra recently observed that if s 0 0, then s 0 1=n with equality holding if and only if G = N is a Frobenius group of order n(n Gamma 1) with n a prime power. Our purpose here is to find the next possible ....

....the notions discussed earlier. We have a complete generalization of Theorem 1.3 only for those pair (G; N) which comes from a covering of connected smooth projective curves with a totally ramified F q rational point. First we note the following easy result (which is essentially proved in [FGS, x13]) We give a different proof suggested by Muller. Lemma 3.1. Let G act on a finite set X. Let OE denote the following permutation character: OE(g) jX g j. Let r = r(X) be the number of common G; N orbits on X. Then (1=jN j) X g2xN OE(g) r: Proof. Clearly, we may assume that G is ....

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M. Fried, R. Guralnick and J. Saxl, Schur covers and Carlitz's conjecture, Israel J. Math. 82 (1993), 157-225.

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